Abstract: We present a novel methodology for designing output-feedback backstepping
boundary controllers for an unstable 1-D diffusion-reaction partial
differential equation with spatially-varying reaction. Using "folding"
transforms the parabolic PDE into a 2X2 coupled parabolic PDE system with
coupling via folding boundary conditions. The folding approach is novel in the
sense that the design of bilateral controllers can be generalized to centering
around arbitrary points, which admit additional design parameters for both the
state-feedback controller and the state observer. The design can be selectively
biased to achieve different performance indicies (e.g. energy, boundedness,
etc). A first backstepping transformation is designed to map the unstable
system into a strict-feedback intermediate target system. A second backstepping
transformation is designed to stabilize the intermediate target system. The
invertibility of the two transformations guarantees that the derived
state-feedback controllers exponentially stabilize the trivial solution of the
parabolic PDE system in the L^2 norm sense. A complementary state observer is
likewise designed for the dual problem, where two collocated measurements are
considered at an arbitrary point in the interior of the domain. The observer
generates state estimates which converge to the true state exponentially fast
in the L^2 sense. Finally, the output feedback control law is formulated by
composing the state-feedback controller with the state estimates from the
observer, and the resulting dynamic feedback is shown to stabilize the trivial
solution of the interconnected system in the L^2 norm sense. Some analysis on
how the selection of these points affect the responses of the controller and
observer are discussed, with simulations illustrating various choices of
folding points and their effect on the stabilization.